Step 1 :Let's denote the number of fish as F, the number of cats as C, and the number of dogs as D. We know that:
Step 2 :\(F + C + D = 349\) (the total number of pets)
Step 3 :\(F = C + D + 19\) (the number of fish is 19 million more than the total number of cats and dogs)
Step 4 :\(C = D + 15\) (15 million more cats are owned than dogs)
Step 5 :We can substitute the second equation into the first one to get an equation with only cats and dogs: \(2C + 2D + 19 = 349\)
Step 6 :Then, we can substitute the third equation into this new equation to get an equation with only dogs: \(4D + 49 = 349\)
Step 7 :Solving this equation will give us the number of dogs: \(D = 75\)
Step 8 :We can substitute the number of dogs into the third equation to get the number of cats: \(C = 90\)
Step 9 :Finally, we can substitute the number of cats and dogs into the first equation to get the number of fish: \(F = 184\)
Step 10 :Final Answer: The number of fish owned by people in this country is \(\boxed{184}\) million, the number of cats owned by people is \(\boxed{90}\) million, the number of dogs owned by people is \(\boxed{75}\) million.